Dot Product of Two Vectors: Formula, Properties and Solved Examples

By Dr. Muhammad Amjad Khadim (PhD Mathematics)

The dot product (also called the scalar product) is the first operation every student meets in vector algebra. It takes two vectors and returns a single number, and that number tells you how much the two vectors point in the same direction. In this guide you will learn the formula, its key properties, and how to use it to find angles, test perpendicular vectors and calculate work done.

Dot Product Formula

For two vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, there are two equivalent definitions:

Component form: a · b = a₁b₁ + a₂b₂ + a₃b₃
Geometric form: a · b = |a| |b| cos θ, where θ is the angle between the vectors (0° ≤ θ ≤ 180°)

Because both forms give the same number, you can combine them to find the angle between any two vectors.

Properties of the Dot Product

Commutative: a · b = b · a
Distributive: a · (b + c) = a · b + a · c
Scalar multiples: (k a) · b = k (a · b)
Self dot product: a · a = |a|²
Unit vectors: i · i = j · j = k · k = 1 and i · j = j · k = k · i = 0
Perpendicular test: if a and b are non-zero, a · b = 0 exactly when a ⟂ b

What the Sign Tells You

a · b > 0 → the angle is acute (less than 90°)
a · b = 0 → the vectors are perpendicular (exactly 90°)
a · b < 0 → the angle is obtuse (greater than 90°)

Solved Examples

Example 1: Are the vectors perpendicular?

Let a = 2i + 3j − k and b = 4i − j + 5k.

a · b = (2)(4) + (3)(−1) + (−1)(5) = 8 − 3 − 5 = 0

Since the dot product is zero, the two vectors are perpendicular.

Example 2: Find the angle between two vectors

Let a = i + 2j + 2k and b = 3i + 4k.

a · b = (1)(3) + (2)(0) + (2)(4) = 11
|a| = √(1 + 4 + 4) = 3 and |b| = √(9 + 0 + 16) = 5
cos θ = 11 / 15 ≈ 0.7333, so θ ≈ 42.83°

Need the answer in radians? Use the degree to radian converter on PakMath (42.83° ≈ 0.7476 rad). For square roots of larger magnitudes, the Adzita square root calculator saves time.

Example 3: Work done by a force

In physics, work is the dot product of force and displacement: W = F · d. If F = 3i + 4j newtons moves an object through d = 5i metres:

W = (3)(5) + (4)(0) + (0)(0) = 15 joules.

Only the part of the force along the displacement does work — the 4j component contributes nothing.

Projection of One Vector on Another

The scalar projection of a on b is (a · b) / |b|. In Example 2, the projection of a on b is 11 / 5 = 2.2 units. The vector projection is ((a · b) / |b|²) b = (11/25)(3i + 4k).

Common Mistakes to Avoid

Writing the answer as a vector — the dot product is always a scalar.
Forgetting negative signs when multiplying components.
Setting your calculator to the wrong angle mode. If you work with cos⁻¹, check degree/radian mode on a scientific calculator.

Practice and Further Reading

Test yourself with our 50 Vector Algebra MCQs with answers, which include many dot product questions. Since the dot product uses cos θ, a quick review of trigonometric identities also helps.

Frequently Asked Questions

Is the dot product a vector or a scalar?

It is a scalar. That is why it is also called the scalar product.

Can the dot product be negative?

Yes. A negative dot product means the angle between the vectors is greater than 90°.

What is the dot product of a vector with itself?

It equals the square of its magnitude: a · a = |a|².

Related: the cross product of two vectors turns two vectors into a new vector perpendicular to both, and you can check any answer with our free vector analysis calculators. Explore more free tools at PakMath calculators and Adzita math calculators.